Question

If \( A \) and \( B \) are square matrices of the same order, then \( (A B^T - B A^T) \) is a:

• A. Symmetric matrix
• B. Skew-symmetric matrix
• C. Null matrix
• D. Unit matrix

Hint:

To check if a matrix is skew-symmetric, verify if its transpose is equal to the negative of the original matrix.

Answer 1

The Correct Option is B

To address the problem, we must examine the nature of the matrix expression \( AB^T - BA^T \) for square matrices \( A \) and \( B \) of the same order.

1. Transposing the Expression:
We calculate the transpose of \( AB^T - BA^T \):

\( (AB^T - BA^T)^T = (AB^T)^T - (BA^T)^T \)
Applying the identity \( (XY)^T = Y^T X^T \), we obtain:

\( (AB^T)^T = B A^T \) and \( (BA^T)^T = A B^T \)
Therefore:
\( (AB^T - BA^T)^T = B A^T - A B^T = - (AB^T - BA^T) \)

2. Interpretation:
As the transpose of the expression is equivalent to its negative, it fulfills the definition of a skew-symmetric matrix:

\( M^T = -M \Rightarrow M \) is skew-symmetric

3. Conclusion:
The matrix \( AB^T - BA^T \) is classified as skew-symmetric.

Final Answer:
The correct classification is (B) skew-symmetric matrix.